function res = exact_solution1(N,mu,x_max,a,num)
h = x_max/N;
k = -mu*h;
T = k*num;
res = zeros(1,N+1);
for i = 1:N+1
    x = mod((i-1)*h+T,2*pi);
    res(i) = initial_function1(x)*exp(-a*T);
end